Additive number theory and the ring of quantum integers

TL;DR

This paper explores the algebraic structure of quantum integers, demonstrating how their addition and multiplication mirror elementary interval decompositions in additive number theory, thus connecting quantum algebra with classical number theory concepts.

Contribution

It introduces a natural polynomial addition and multiplication for quantum integers that form a ring and field, linking quantum algebra to additive number theory.

Findings

Quantum integers form a ring and a field with polynomial operations.

Quantum integer operations correspond to elementary interval decompositions.

The work bridges quantum algebra and additive number theory.

Abstract

Let $m$ and $n$ be positive integers. For the quantum integer $[n]_q = 1 + q + ... + q^{n-1}$ there is a natural polynomial addition such that $[m]_q \oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that $[m]_q \otimes_q [n]_q = [mn]_q$. These constructions lead to the construction of the ring of quantum integers and the field of quantum rational numbers. It is also shown that addition and multiplication of quantum integers are equivalent to elementary decompositions of intervals of integers in additive number theory.